Fractals/Fibonacci Project: Reflection

Camille Favreau

Dan Ryska

Fractals Reflection

How do Fibonacci numbers show up in nature?
How are Fibonacci's represented in your piece?
In general, describe what is going on in your piece.



On many plants, the number of petals is a Fibonacci number and correspond to the sequence:
buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
My piece is a collage of representation of Fibonacci numbers in flowers; the Fibonacci spiral in roses, in fruit and vegetables (Romanesque Broccoli/Cauliflower (or Romanesco). This is an amazing thing that nature evolved in this such ‘organized way’. It can lead us to the question how does it work? Why nature is this such organized  chaos?

Connection:
The connection between nature and mathematics is clearly present in my piece. As said above, the innate organization is related to the mathematics sequences discovered.

The Golden Ratio

• In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.
• The Golden ratio is a special ratio. Phi (lowercase) is used to refer to this ratio, and the value is , which is approximately 1.618.
  The uppercase phi () is used for the reciprocal of the golden ratio, which is 1/.
  

• There are many different ways in which the Fibonacci sequence is related to the Golden Ratio. Firstly, the further you go to the right of the sequence, the more the ratio of one term to the one before it estimates the Golden Ratio. The table below shows the first few numbers and their ratios:

First number	Second number	Ratio
0	1	-
1	1	1.0000
1	2	2.0000
2	3	1.5000
3	5	1.6667
5	8	1.6000
8	13	1.6250
13	21	1.6154
21	34	1.6190
34	55	1.6176
55	89	1.6182
89	144	1.6180

 

• The golden ratio also determines how attractive a person is.
  For example, the American pop singer and actress Jessica Simpson is attractive because the proportion of her face fits geometrically on the human face mask which conforms to some aspect of the Golden Ratio.
  

• Do you think the Golden Ratio "naturally" occurs in nature and art/architecture or do people "force" it to be there?

Obviously, nature doesn't know about the golden ratio, but for art and architecture, which are created by people, in order to look "good" and well proportioned, yes they "force" it to be there.

Facts about Fibonacci

1,1,2,3,5,8, 13, 21, 34, ...,

un represents the nth term of the Fibonacci sequence, show that

u

n =

1

+ 5 n ? 1 ? 5 n

2

., n 5n = 1, 2, 3, ...x, y, x + y, ...

Petals on flowers

Probably most of us have never taken the time to examine very carefully the number or arrangement of petals on a flower. If we were to do so, we would find that the number of petals on a flower, that still has all of its petals intact and has not lost any, for many flowers is a Fibonacci number: 

• 3 petals: lily, iris
• 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
• 8 petals: delphiniums
• 13 petals: ragwort, corn marigold, cineraria,
• 21 petals: aster, black-eyed susan, chicory
• 34 petals: plantain, pyrethrum
• 55, 89 petals: michaelmas daisies, the asteraceae family

http://www.world-mysteries.com/sci_17.htm

If

"I'm not smart, but I like to observe. Millions saw the apple fall, but Newton was the one who asked why." - Baruch

"I'm not smart, but I like to observe. Millions saw the apple fall, but Newton was the one who asked why."     - Baruch

Observation takes time, concentration and curiosity. After observing the siuation, everything becomes clear, at least more than it was. Taking the time to see, taking the time to appreciate, taking the time to think, taking the time to live, that makes evrything having more sense.
You don't have to be smart to be logical; logical mind wait to observe all the pieces of the situation, before acting.
A lot of us live their lives, as they can, without asking and understanding why life is as it is, that's why people like Newton became famous, they were able to ask the good existentials questions of life.